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EEE 498/598 Overview of Electrical Engineering

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## EEE 498/598 Overview of Electrical Engineering

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**EEE 498/598Overview of Electrical Engineering**Lecture 2: Introduction to Electromagnetic Fields; Maxwell’s Equations; Electromagnetic Fields in Materials; Phasor Concepts; Electrostatics: Coulomb’s Law, Electric Field, Discrete and Continuous Charge Distributions; Electrostatic Potential 1**Lecture 2 Objectives**• To provide an overview of classical electromagnetics, Maxwell’s equations, electromagnetic fields in materials, and phasor concepts. • To begin our study of electrostatics with Coulomb’s law; definition of electric field; computation of electric field from discrete and continuous charge distributions; and scalar electric potential. 2**Introduction to Electromagnetic Fields**• Electromagnetics is the study of the effect of charges at rest and charges in motion. • Some special cases of electromagnetics: • Electrostatics: charges at rest • Magnetostatics: charges in steady motion (DC) • Electromagnetic waves: waves excited by charges in time-varying motion 3**Circuit**Theory Introduction to Electromagnetic Fields Maxwell’s equations Fundamental laws of classical electromagnetics Geometric Optics Electro-statics Magneto-statics Electro-magnetic waves Special cases Statics: Transmission Line Theory Input from other disciplines Kirchoff’s Laws 4**Introduction to Electromagnetic Fields**• transmitter and receiver • are connected by a “field.” 5**2**3 1 4 Introduction to Electromagnetic Fields High-speed, high-density digital circuits: • consider an interconnect between points “1” and “2” 6**Introduction to Electromagnetic Fields**• Propagation delay • Electromagnetic coupling • Substrate modes 7**Introduction to Electromagnetic Fields**• When an event in one place has an effect on something at a different location, we talk about the events as being connected by a “field”. • A fieldis a spatial distribution of a quantity; in general, it can be either scalar or vector in nature. 8**Introduction to Electromagnetic Fields**• Electric and magnetic fields: • Are vector fields with three spatial components. • Vary as a function of position in 3D space as well as time. • Are governed by partial differential equations derived from Maxwell’s equations. 9**Introduction to Electromagnetic Fields**• A scalaris a quantity having only an amplitude (and possibly phase). • A vectoris a quantity having direction in addition to amplitude (and possibly phase). Examples: voltage, current, charge, energy, temperature Examples: velocity, acceleration, force 10**Introduction to Electromagnetic Fields**• Fundamental vector field quantities in electromagnetics: • Electric field intensity • Electric flux density (electric displacement) • Magnetic field intensity • Magnetic flux density units = volts per meter (V/m = kg m/A/s3) units = coulombs per square meter (C/m2 = A s /m2) units = amps per meter (A/m) units = teslas = webers per square meter (T = Wb/ m2 = kg/A/s3) 11**Introduction to Electromagnetic Fields**• Universal constants in electromagnetics: • Velocity of an electromagnetic wave (e.g., light) in free space (perfect vacuum) • Permeability of free space • Permittivity of free space: • Intrinsic impedance of free space: 12**Introduction to Electromagnetic Fields**• Relationships involving the universal constants: In free space: 13**sources**Ji, Ki fields E, H Introduction to Electromagnetic Fields • Obtained • by assumption • from solution to IE Solution to Maxwell’s equations Observable quantities 14**Maxwell’s Equations**• Maxwell’s equations in integral form are the fundamental postulates of classical electromagnetics - all classical electromagnetic phenomena are explained by these equations. • Electromagnetic phenomena include electrostatics, magnetostatics, electromagnetostatics and electromagnetic wave propagation. • The differential equations and boundary conditions that we use to formulate and solve EM problems are all derived from Maxwell’s equations in integral form. 15**Maxwell’s Equations**• Various equivalence principles consistent with Maxwell’s equations allow us to replace more complicated electric current and charge distributions with equivalent magnetic sources. • These equivalent magnetic sources can be treated by a generalization of Maxwell’s equations. 16**Maxwell’s Equations in Integral Form (Generalized to**Include Equivalent Magnetic Sources) Adding the fictitious magnetic source terms is equivalent to living in a universe where magnetic monopoles (charges) exist. 17**Continuity Equation in Integral Form (Generalized to Include**Equivalent Magnetic Sources) • The continuity equations are implicit in Maxwell’s equations. 18**S**C dS S V dS Contour, Surface and Volume Conventions • open surface S bounded by • closed contour C • dS in direction given by • RH rule • volume V bounded by • closed surface S • dS in direction outward • from V 19**Electric Current and Charge Densities**• Jc = (electric) conduction current density (A/m2) • Ji = (electric) impressed current density (A/m2) • qev = (electric) charge density (C/m3) 20**Magnetic Current and Charge Densities**• Kc = magnetic conduction current density (V/m2) • Ki = magnetic impressed current density (V/m2) • qmv = magnetic charge density (Wb/m3) 21**Maxwell’s Equations - Sources and Responses**• Sources of EM field: • Ki, Ji, qev, qmv • Responses to EM field: • E, H, D, B, Jc, Kc 22**Maxwell’s Equations in Differential Form (Generalized to**Include Equivalent Magnetic Sources) 23**Continuity Equation in Differential Form (Generalized to**Include Equivalent Magnetic Sources) • The continuity equations are implicit in Maxwell’s equations. 24**Electromagnetic Boundary Conditions**Region 1 Region 2 25**Surface Current and Charge Densities**• Can be either sources of or responses to EM field. • Units: • Ks - V/m • Js - A/m • qes - C/m2 • qms - W/m2 27**Electromagnetic Fields in Materials**• In time-varying electromagnetics, we consider E and H to be the “primary” responses, and attempt to write the “secondary” responses D, B, Jc, and Kcin terms of E and H. • The relationships between the “primary” and “secondary” responses depends on the medium in which the field exists. • The relationships between the “primary” and “secondary” responses are called constitutive relationships. 28**Electromagnetic Fields in Materials**• Most general constitutive relationships: 29**Electromagnetic Fields in Materials**• In free space, we have: 30**Electromagnetic Fields in Materials**• In a simple medium, we have: • linear(independent of field strength) • isotropic(independent of position within the medium) • homogeneous(independent of direction) • time-invariant(independent of time) • non-dispersive (independent of frequency) 31**Electromagnetic Fields in Materials**• e = permittivity = ere0 (F/m) • m = permeability = mrm0(H/m) • s = electric conductivity = ere0(S/m) • sm = magnetic conductivity = ere0(W/m) 32**Phasor Representation of a Time-Harmonic Field**• A phasor is a complex number representing the amplitude and phase of a sinusoid of known frequency. phasor frequency domain time domain 33**Phasor Representation of a Time-Harmonic Field**• Phasors are an extremely important concept in the study of classical electromagnetics, circuit theory, and communications systems. • Maxwell’s equations in simple media, circuits comprising linear devices, and many components of communications systems can all be represented as linear time-invariant (LTI) systems. (Formal definition of these later in the course …) • The eigenfunctions of any LTI system are the complex exponentials of the form: 34**If the input to an LTI system is a sinusoid of frequency w,**then the output is also a sinusoid of frequency w (with different amplitude and phase). Phasor Representation of a Time-Harmonic Field A complex constant (for fixed w); as a function of w gives the frequency response of the LTI system. 35**Phasor Representation of a Time-Harmonic Field**• The amplitude and phase of a sinusoidal function can also depend on position, and the sinusoid can also be a vector function: 36**Phasor Representation of a Time-Harmonic Field**• Given the phasor (frequency-domain) representation of a time-harmonic vector field, the time-domain representation of the vector field is obtained using the recipe: 37**Phasor Representation of a Time-Harmonic Field**• Phasors can be used provided all of the media in the problem are linear no frequency conversion. • When phasors are used, integro-differential operators in time become algebraic operations in frequency, e.g.: 38**Time-Harmonic Maxwell’s Equations**• If the sources are time-harmonic (sinusoidal), and all media are linear, then the electromagnetic fields are sinusoids of the same frequency as the sources. • In this case, we can simplify matters by using Maxwell’s equations in the frequency-domain. • Maxwell’s equations in the frequency-domain are relationships between the phasor representations of the fields. 39**Maxwell’s Equations in Differential Form for Time-Harmonic**Fields 40**Maxwell’s Equations in Differential Form for Time-Harmonic**Fields in Simple Medium 41**Circuit**Theory Electrostatics as a Special Case of Electromagnetics Maxwell’s equations Fundamental laws of classical electromagnetics Geometric Optics Electro-statics Magneto-statics Electro-magnetic waves Special cases Statics: Transmission Line Theory Input from other disciplines Kirchoff’s Laws 42**Electrostatics**• Electrostaticsis the branch of electromagnetics dealing with the effects of electric charges at rest. • The fundamental law of electrostatics is Coulomb’s law. 43**Electric Charge**• Electrical phenomena caused by friction are part of our everyday lives, and can be understood in terms of electrical charge. • The effects of electrical charge can be observed in the attraction/repulsion of various objects when “charged.” • Charge comes in two varieties called “positive” and “negative.” 44**Electric Charge**• Objects carrying a net positive charge attract those carrying a net negative charge and repel those carrying a net positive charge. • Objects carrying a net negative charge attract those carrying a net positive charge and repel those carrying a net negative charge. • On an atomic scale, electrons are negatively charged and nuclei are positively charged. 45**Electric Charge**• Electric charge is inherently quantized such that the charge on any object is an integer multiple of the smallest unit of charge which is the magnitude of the electron charge e= 1.602 10-19C. • On the macroscopic level, we can assume that charge is “continuous.” 46**Coulomb’s Law**• Coulomb’s law is the “law of action” between charged bodies. • Coulomb’s law gives the electric force between two point chargesin an otherwise empty universe. • A point charge is a charge that occupies a region of space which is negligibly small compared to the distance between the point charge and any other object. 47**Coulomb’s Law**Q1 Q2 Unit vector in direction of R12 Force due to Q1 acting on Q2 48**Coulomb’s Law**• The force on Q1due to Q2 is equal in magnitude but opposite in direction to the force on Q2 due to Q1. 49**Qt**Q Electric Field • Consider a point charge Q placed at the origin of a coordinate system in an otherwise empty universe. • A test charge Qtbrought near Qexperiences a force: 50